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The matrix exponentiation technique has been used before to solve ordinary three-term recurrence relations, such as for this example here.

I now have a matrix recurrence relation such as \begin{equation} R_{n+2}+A_n R_{n+1} + B_n R_{n}=0 \end{equation} for $n\ge 0$ and some initial conditions $R_0$ and $R_1$ and $A_n$ and $B_n$ are invertible 2x2 matrices which depend on $n$ polynomically. Would similar techniques work here?

Alternatively, one can try to use generating functions, but because $A_n$ and $B_n$ depend polynomially on $n$ (up to $n^2$), those are nontrivial to solve and it is unclear if the resulting matrix ODE would be even solvable.

J.Agusti
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If I understand the goal correctly you need to create blocks: $$J(n) = \begin{bmatrix} R(n) \\ R(n+1) \end{bmatrix} $$

(Recall that matrix multiplication is recursive (ex: 4x4 marix is just a 2x2 of 2x2 blocks etc...))

Then the equation $R(n+2) +A_n R(n+1) +B_n R(n) = 0$ can be rewritten as

$$ \begin{bmatrix} R(n+1) \\ R(n+2) \end{bmatrix} + \begin{bmatrix} 0 && -I_n \\ B_n && A_n \end{bmatrix} \begin{bmatrix} R(n) \\ R(n+1) \end{bmatrix} = 0 $$

I.E.

$$ J(n+1) = -\begin{bmatrix} 0 && -I_n \\ B_n && A_n \end{bmatrix} J(n) $$

At this point things get a little harder. You are dealing with the matrix equivalent of say $f(n+1) = n*f(n)$ whereas to use matrix exponentials you want the matrix equivalent of $f(n+1) = 2*f(n)$. The latter being a constant leading to a nice matrix formula but the former being a varying multiple.

I'll have to return to this question but one basically wants to ask two questions to wrap up here:

  1. Given $f(n+1) = P(n) * f(n)$ how does one interpolate the curve itself? (ex finding the gamma function from the factorial relation)

  2. Then how does (1) generalize to the matrix setting?

I'm not sure off the top of my head how to proceed with these. Although for (1) you can probably modify @JM's techniques for the partition function here

The gist of his technique is to to consider $f(x) = \sum_{n=0}^{\infty} P(n) x^n$ and then to evaluate the fractional derivative $\frac{1}{\Gamma({\alpha +1})}\frac{d^{\alpha}}{dx^{\alpha}}[f(x)]_{\text{evaluate at} \ x = 0}$ for a suitable choice of contour to create a curve that interpolates $P(n)$ and is "natural" in a vague sense (JM found considering a sequence of circles approaching radius 1 from 0 to be the natural "interpolater" for the partition function and that might hold for many other functions).

Now we want to put this into the matrix setting. Which can be done by letting $x$ in $f(x)$ above no longer be a complex number but a matrix. The only caveat here is "what is a natural contour in the matrix setting?" and more importantly "is there a natural contour?".

In the event we weren't dealing with shift $J(n+1) = A(n) J(n)$ but a derivative $J' = A(x)J(x)$ then the Time ordered exponential is what you seek. So you are essentially trying to find a "discrete time ordered exponential".

*you mention that $A_n, B_n$ are polynomial. If they were merely linear or even affine functions of $n$ then techniques involving the matrix gamma function might be useful.

Sidharth Ghoshal
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    Thanks for the derivation! I am now a bit unsure if we can even use the matrix exponential for non-constant coefficients. I realize this by your last formula: as $J(k)=(-1)^k M_{n=0}^{k} J(0) $, where $M_{n}$ is the 4x4 matrix. Basically, we just evaluate $M_{n}$ at $n=0$ at that feels not correct. – J.Agusti May 28 '25 at 15:38
  • Going from the $J(n+1) = \cdots$ line to the next isn’t correct. – A rural reader May 28 '25 at 15:42
  • OH SHUCKS, I completely forgot $A_n$ And $B_n$ vary as a function of $n$! thanks for pointing that out @Aruralreader. Yea unfortunately this is considerably trickier – Sidharth Ghoshal May 28 '25 at 15:52
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    J.Augusti, I have given some vague directions about how to go from varying matrix to a continuous curve. They are probably quite hard to do but people have gone on similar journeys before so its not hopeless. – Sidharth Ghoshal May 28 '25 at 16:04
  • Thanks for the extended answer! Unfortunately, I found it a bit hard to apply to my problem. The form of the matrix elements is polynomials as $\frac{n^2+n+\lambda}{(1+2n)^2}$ or similar, so they cannot be linearized for the gamma function. I should also have mentioned that I am interested in the asymptotic limit $n\rightarrow \infty$, which I don't know if this changes anything. I proceed with generating functions to obtain a matrix ODE, maybe it helps. – J.Agusti May 28 '25 at 17:03
  • Ok so it sounds to me like you have a particular infinite matrix product (of varying matrices) that you want to evaluate (if it converges) and if it doesn't converge then you want to be able to evaluate a divergent regularization of it. Is this correct? And if so, I might have some useful tools as I do enjoy working with divergent series. – Sidharth Ghoshal May 28 '25 at 19:11